Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173

 
 

 

Metric and geometric quasiconformality in Ahlfors regular Loewner spaces


Author: Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 5 (2001), 21-73
MSC (2000): Primary 30C65; Secondary 28A78, 46E35, 43A85
DOI: https://doi.org/10.1090/S1088-4173-01-00064-9
Published electronically: August 8, 2001
MathSciNet review: 1872156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu's generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension $Q>1$, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.

We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper $Q$-regular Loewner space, $Q>1$, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.


References [Enhancements On Off] (What's this?)

  • 1. P. Alestalo and J. Väisälä, Quasisymmetric embeddings of products of cells into the Euclidean space, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 375-392. MR 95h:30023
  • 2. Z. Balogh, Hausdorff dimension distortion of QC mappings on the Heisenberg group, J. Anal. Math. 83 (2001), 289-312.
  • 3. Z. Balogh and P. Koskela, Quasiconformality, quasisymmetry and removability in Loewner spaces, Duke Math. J. 101 (2000), no. 3, 554-577, with an appendix by J. Väisälä. MR 2001d:30029
  • 4. Z. Balogh and J. T. Tyson, Polar coordinates and regularity of quasiconformal mappings in Carnot groups, preprint, 2001.
  • 5. Z. M. Balogh, I. Holopainen, and J. T. Tyson, Singular solutions, homogeneous norms and quasiconformal mappings on Carnot groups, Preprint #269, University of Helsinki, September 2000.
  • 6. N. Benakli, Polyèdres hyperboliques, passage du local au global, Thèse, Université Paris-Sud, 1992.
  • 7. C. J. Bishop and J. T. Tyson, Locally minimal sets for conformal dimension, Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 361-373.
  • 8. B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Lecture Notes in Mathematics, no. 1351, Springer-Verlag, Berlin, 1988, pp. 52-68. MR 90b:46068
  • 9. M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), 245-268. MR 98c:20056
  • 10. M. Bourdon and H. Pajot, Poincaré inequalities and quasiconformal structure on the boundaries of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315-2324. MR 99j:30024
  • 11. A. Bruckner, J. Bruckner, and B. Thomson, Real analysis, Prentice-Hall, N.J., 1997.
  • 12. J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. MR 95k:30046
  • 13. J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math., no. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 95g:20045
  • 14. J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension $3$, Trans. Amer. Math. Soc. 350 (1998), 809-849. MR 98i:57023
  • 15. L. Capogna, Regularity of quasilinear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867-889.
  • 16. -, Regularity for quasilinear equations and $1$-quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263-295. MR 2000a:35027
  • 17. L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Man. Math. 86 (1995), 267-281. MR 96f:30019
  • 18. C. Champetier, Propriétés statistiques des groupes de présentation finie, Adv. Math. 116 (1995), 197-262. MR 96m:20056
  • 19. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. MR 2000g:53043
  • 20. W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98-105. MR 1:313d
  • 21. M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, no. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 92f:57003
  • 22. G. David and S. Semmes, Fractured fractals and broken dreams: self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, vol. 7, Clarendon Press Oxford University Press, 1997. MR 99h:28018
  • 23. G. A. Edgar, Packing measure as a gauge variation, Proc. Amer. Math. Soc. 122 (1994), 167-174. MR 94k:28009
  • 24. -, Integral, probability, and fractal measures, Springer-Verlag, New York, 1998. MR 99c:28024
  • 25. -, Packing measure in general metric spaces, Real Anal. Exchange (to appear).
  • 26. H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer-Verlag, New York, 1969. MR 41:1976
  • 27. B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-219. MR 20:4187
  • 28. F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499-519. MR 24:A2677
  • 29. -, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 25:3166
  • 30. -, The ${L}\sp{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 53:5861
  • 31. -, Quasiconformal mappings, pp. 213-268, Internat. Atomic Energy Agency, Vienna, 1976. MR 58:1144
  • 32. P. Haj\lasz and P. Koskela, Sobolev met Poincaré, Memoirs Amer. Math. Soc. 145 (2000), no. 688, 101 pp. MR 2000j:46063
  • 33. B. Hanson and J. Heinonen, An $n$-dimensional space that admits a Poincaré inequality but has no manifold points, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3379-3390. MR 2001e:43004
  • 34. J. Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994), vol. 68, Ber. Univ. Jyväskylä Math. Inst., Jyväskylä, 1995, pp. 1-31. MR 96j:22015
  • 35. -, A capacity estimate on Carnot groups, Bull. Sci. Math. 119 (1995), 475-484. MR 96j:22011
  • 36. J. Heinonen and I. Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal. 7 (1997), no. 1, 109-148. MR 99i:30037
  • 37. J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61-79. MR 96e:30051
  • 38. -, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 99j:30025
  • 39. J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87-139.
  • 40. J. Heinonen and S. Semmes, Thirty-three yes or no questions about mappings, measures, and metrics, Conform. Geom. Dyn. 1 (1997), 1-12. MR 99h:28012
  • 41. W. Hurewicz and H. Wallman, Dimension theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 3:312b
  • 42. A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309-338. MR 86m:32035
  • 43. -, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), 1-87. MR 96c:30021
  • 44. P. Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291-304. MR 2001g:46077
  • 45. P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1-17. MR 99e:46042
  • 46. T. Laakso, Ahlfors ${Q}$-regular spaces with arbitrary ${Q}$ admitting weak Poincaré inequalities, Geom. Funct. Anal. 10 (2000), no. 1, 111-123.
  • 47. C. Loewner, On the conformal capacity in space, J. Math. Mech. 8 (1959), 411-414. MR 21:3538
  • 48. G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5 (1995), no. 2, 402-433. MR 96m:53038
  • 49. P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 96h:28006
  • 50. G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 38:4679
  • 51. P. Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. MR 90k:53079
  • 52. -, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60. MR 90e:53058
  • 53. P. K. Rashevsky, Any two points of a totally nonholonomic space may be connected by an admissible line, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math. 2 (1938), 83-94, in Russian.
  • 54. H. M. Reimann, Rigidity of ${H}$-type groups, Math. Z. (to appear).
  • 55. -, An estimate for pseudoconformal capacities on the sphere, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 315-324. MR 90m:32041
  • 56. H. M. Reimann and F. Ricci, The complexified Heisenberg group, Preprint, 2000.
  • 57. Yu. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), 657-675. MR 98h:46031
  • 58. C. A. Rogers, Hausdorff measures, Cambridge University Press, London, 1970. MR 43:7576
  • 59. W. Rudin, Real and complex analysis, third ed., McGraw-Hill, New York, 1987. MR 88k:00002
  • 60. X. Saint Raymond and C. Tricot, Packing regularity of sets in $n$-space, Math. Proc. Cambridge Philos. Soc. 103 (1988), 133-145. MR 88m:28002
  • 61. S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. 2 (1996), 155-295. MR 97j:46033
  • 62. N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243-279. CMP 2001:07
  • 63. P. Tang, Quasiconformal homeomorphisms on CR $3$-manifolds with symmetries, Math. Z. 219 (1995), 49-69. MR 96i:32016
  • 64. S. J. Taylor and C. Tricot, The packing measure of rectifiable subsets of the plane, Math. Proc. Cambridge Philos. Soc. 99 (1986), 285-296. MR 87b:28008
  • 65. B. S. Thomson, Construction of measures in metric spaces, J. London Math. Soc. (2) 14 (1976), 21-24. MR 54:10537
  • 66. C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. MR 84d:28013
  • 67. P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160. MR 83b:30019
  • 68. P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. MR 82g:30038
  • 69. J. T. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525-548. MR 99i:30038
  • 70. -, Analytic properties of locally quasisymmetric mappings from Euclidean domains, Indiana Univ. Math. J. 49 (2000), no. 3, 995-1016. CMP 2001:06
  • 71. J. Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, no. 229, Springer-Verlag, Berlin, 1971. MR 56:12260
  • 72. -, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), 191-204. MR 82i:30031
  • 73. -, Quasiconformal maps of cylindrical domains, Acta Math. 162 (1989), 201-225. MR 90f:30034
  • 74. S. Willard, General topology, Addison-Wesley, Reading, MA, 1970. MR 41:9173
  • 75. W. P. Ziemer, Extremal length and $p$-capacity, Michigan Math. J. 16 (1969), 43-51. MR 40:346

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30C65, 28A78, 46E35, 43A85

Retrieve articles in all journals with MSC (2000): 30C65, 28A78, 46E35, 43A85


Additional Information

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
Email: tyson@math.sunysb.edu

DOI: https://doi.org/10.1090/S1088-4173-01-00064-9
Keywords: Quasiconformal/quasisymmetric map, conformal modulus, Loewner condition, Hausdorff/packing measure, Poincar\'{e} inequality
Received by editor(s): May 31, 2000
Received by editor(s) in revised form: June 4, 2001
Published electronically: August 8, 2001
Additional Notes: The results of this paper form part of the author’s Ph.D. thesis completed at the University of Michigan in 1999. Research supported by an NSF Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society